Why does the parent go haywire, when the child's rigid body physics is set to "animated"?
I still don't know why it does that, but I have found a way to perfectly avoid this bug. 

Suprisingly, is has nothing to do with the fracture modifier, or the child, but instead with the rigid body simulation.

In order to avoid this bug, 2 small steps are required:
1) After simulating the rigid-body-physics, the simulation must be baked.
The settings for that are not in the "Physics" tab, but in the "Scene" tab.

2) After baking, the simulation must be "baked to keyframes"
The settings for that aren't in the "physics" tab nor in the "scene" tab, but in a place entirely else. Both tabs are in the Editor-type "Properties".
But all the way on the other side in the so called "3D View Tools pane", in the "Physics tab" are the settings needed.



After these 2 steps are done, the rigid body settings in the "physics" tab can be closed, because the simulation is now an animation. The fracture modifier and the rigid-body phsyics on all children now work the way they schould.